Geodesic
Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 123-128

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Let a function $f$ be holomorphic in the unit ball $\mathbb B^n$, continuous in the closed ball $\overline{\mathbb B}^n$, and let $f(z)\ne0$, $z\in\mathbb B^n$. Assume that $|f|$ belongs to the $\alpha$-Hölder class on the unit sphere $S^n$, $0\alpha\leq1$. The present paper is devoted to the proof of statement that $f$ belongs to the $\alpha/2$-Hölder class on $\overline{\mathbb B}^n$. 
@article{ZNSL_2016_447_a8,
     author = {N. A. Shirokov},
     title = {Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {123--128},
     publisher = {mathdoc},
     volume = {447},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/}
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N. A. Shirokov. Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 123-128. http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/